Contribution à la modélisation des écoulements diphasiques en milieu poreux et au couplage interfacial

HAL Id: tel-01624467

https://tel.archives-ouvertes.fr/tel-01624467

Submitted on 26 Oct 2017

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

Contribution à la modélisation des écoulements

diphasiques en milieu poreux et au couplage interfacial

Laëtitia Girault

To cite this version:

Laëtitia Girault. Contribution à la modélisation des écoulements diphasiques en milieu poreux et

au couplage interfacial. Equations aux dérivées partielles [math.AP]. Université de Provence (Aix

Marseille 1), 2010. Français. �tel-01624467�

Contribution `

a la mod´elisation

des ´ecoulements diphasiques en milieu poreux

et au couplage interfacial

La¨

etitia GIRAULT

LATP, UMR CNRS 6632, Marseille

Math´

ematiques et Informatique de Marseille, E.D. num´

ero 184

Th`

ese pr´

epar´

ee `

a:

EDF-R&D, MFEE, 6 quai Watier, 78400 Chatou

Jury:

Mme Annalisa Ambroso, Examinateur

M. Fr´

ed´

eric Coquel, Examinateur

M. Thierry Gallou¨

et, Examinateur

Mme Edwige Godlewski, Rapporteur

M. Philippe Helluy, Rapporteur

M. Jean-Marc H´

erard, Directeur de Th`

ese

Mme Raphaele Herbin, Examinateur

June 1, 2010

Contents

1 Introduction 4

2 Comparaison de sch´emas pour la simulation d’´ecoulements diphasiques

bifluides en milieu poreux 14

3 Evaluation de conditions de couplage au niveau de l’interface fluide/poreux et simulations multidimensionnelles 43 4 Couplage interfacial d’un mod`ele triphasique et d’un mod`ele

diphasique 71

5 Conclusion 101

6 Annexe 104

1

Introduction

L’´etude des ´ecoulements multiphasiques intervient dans de nombreuses appli-cations de la m´ecanique des fluides num´erique. Le domaine est vaste car il s’agit d’´etudier des ´ecoulements comprenant diff´erentes phases ayant des pro-pri´et´es thermodynamiques diff´erentes et pouvant interagir ensemble. L’industrie p´etrochimique, par exemple, s’int´eresse `a la mod´elisation d’´ecoulements triphasiques type eau - huile - gaz, et la mod´elisation d’´ecoulements diphasiques de type gaz - particules va trouver des applications aussi bien pour l’´etude d’explosifs que pour des ´etudes environnementales de type d´epˆots de particules. De la mˆeme mani`ere, les ´ecoulements diphasiques eau liquide - vapeur d’eau vont pouvoir ˆetre ´etudi´es dans des domaines aussi vari´es que la mod´elisation du brouillard ou le nucl´eaire civil. C’est dans ce dernier domaine que se situent les travaux pr´esent´es ici.

Mieux comprendre le comportement des fluides au sein des circuits hy-drauliques des centrales est un enjeu important pour la recherche et l’industrie nucl´eaire civile. Pour am´eliorer les performances et le niveau de sˆuret´e des cen-trales, il est en effet n´ecessaire de pouvoir pr´edire des comportements de fluides diphasiques eau liquide - vapeur d’eau, pour ainsi pr´evoir l’´evolution du syst`eme `

a la suite d’un incident d’exploitation ou d’un accident.

Remarque: par la suite, nous parlerons ”d’eau”, ”de vapeur” et d’´ecoulement diphasique ”eau-vapeur”, sans pr´eciser qu’il s’agit d’eau liquide et de vapeur d’eau.

Afin de clarifier le contexte industriel de cette th`ese, et de mettre en ´evidence l’enjeu de la mod´elisation et de la simulation num´erique, il est utile de rappeler quelques concepts sur le fonctionnement d’une centrale nucl´eaire, et les princi-pales situations accidentelles pouvant avoir lieu.

Contexte industriel

Description simplifi´ee d’une centrale nucl´eaire de type REP

Il existe diff´erents mod`eles - ou ”fili`eres” - de centrales nucl´eaires, se car-act´erisant par trois principaux ´el´ements et leur association:

• le combustible : uranium naturel, uranium enrichi, plutonium,

• le mod´erateur (substance utilis´ee pour favoriser le d´eveloppement de la r´eaction de fission en chaˆıne) : eau ordinaire, eau lourde, graphite, • le caloporteur (fluide d’extraction de la chaleur produite par le combustible

car-bonique, sodium, h´elium.

La fili`ere des R´eacteurs `a Eau sous Pression (REP en fran¸cais, ou PWR en anglais pour Pressurized Water Reactor) est la plus r´epandue au monde. Toutes les centrales nucl´eaires fran¸caises en activit´e appartiennent `a cette fili`ere `a eau sous pression (combustible : uranium l´eg`erement enrichi ; mod´erateur et calo-porteur : eau ordinaire sous pression), et sont r´ealis´ees par s´eries standardis´ees correspondant `a diff´erents paliers de puissance ´electrique des r´eacteurs : 900, 1300, 1450 m´egawatts1

Une centrale nucl´eaire de type REP comporte trois circuits hydrauliques principaux ind´ependants (voir figure 1)

Figure 1: Sch´ema de fonctionnement d’une centrale nucl´eaire `a R´eacteur `a Eau sous Pression1

• Le circuit primaire: pour produire de la chaleur

il s’agit d’un circuit ferm´e situ´e dans le bˆatiment r´eacteur dans lequel cir-cule de l’eau maintenue sous une pression d’environ 155 bar pour l’empˆecher de bouillir. En situation normale, ce circuit ne contient que de l’eau sous forme liquide. Un syst`eme de pompes envoie l’eau dans le r´eacteur, o`u la fission des atomes d’uranium produit une grande quantit´e de chaleur. Cette chaleur fait augmenter la temp´erature de l’eau `a plus de 330 degr´es C avant qu’elle rejoigne le g´en´erateur de vapeur, ´echangeur thermique avec le circuit secondaire.

1_{”Comment fonctionne une centrale nucl´eaire”, sur le site internet de la Soci´et´e Fran¸caise}

d’Energie Nucl´eaire http://www.sfen.org/fr/intro/comment.htm - consult´e en septembre 2009

• Le circuit secondaire: pour produire de la vapeur

Au contact des tubes du g´en´erateur de vapeur parcourus par l’eau du cir-cuit primaire, l’eau du circir-cuit secondaire s’´echauffe `a son tour et est ainsi port´ee `a ´ebullition pour produire de la vapeur. Cette vapeur fait tourner une turbine, qui entraˆıne l’alternateur produisant de l’´electricit´e. Apr`es son passage dans la turbine, la vapeur est refroidie, condens´ee dans le con-denseur, et renvoy´ee vers le g´en´erateur de vapeur sous forme d’eau liquide pour un nouveau cycle.

• Le circuit de refroidissement: pour condenser la vapeur et ´evacuer la chaleur

L’´energie extraite dans le condenseur est ´evacu´ee par l’interm´ediaire d’un troisi`eme circuit ind´ependant des deux autres, le circuit de refroidissement. Sa fonction est de condenser la vapeur sortant de la turbine du circuit secondaire en eau. La chaleur est ´echang´ee avec une source froide (eau d’une rivi`ere ou de la mer) et/ou `a l’aide d’une tour de refroidissement ou d’a´eror´efrig´erant. Dans le cas d’un ´echange direct avec la mer, le circuit est ouvert. Dans le cas d’un a´eror´efrig´erant, l’essentiel de cette eau retourne vers le condenseur, une petite partie s’´evapore dans l’atmosph`ere, ce qui provoque ces panaches blancs caract´eristiques des centrales nucl´eaires. Dans une centrale de type REP, seule l’eau du circuit primaire circule dans le coeur du r´eacteur, il est donc imp´eratif de pouvoir d´eterminer ses caract´eristiques. Dans tout le reste de cette th`ese, nous nous int´eresserons uniquement `a la sim-ulation d’´ecoulements fluides au sein du circuit primaire.

La probl´ematique des situations accidentelles :

Bien que le circuit primaire soit ferm´e, ´etanche, et distinct des autres circuits afin d’´eviter toute dispersion de substance radioactive `a l’ext´erieur de la cen-trale en cas d’incident d’exploitation, les ´etudes de sˆuret´e envisagent un certain nombre de situations accidentelles.

Dans le cas hypoth´etique d’une br`eche dans le circuit primaire, une partie de l’eau qui y circule s’en ´echappe. On se retrouve alors dans le cas d’un Acci-dent de Perte de R´efrig´erant Primaire (APRP). La d´epressurisation du circuit primaire entraˆıne une ´ebullition brutale, et la mont´ee des temp´eratures dans le coeur pourraient conduire `a des d´eformations et `a des dommages importants. En effet, la mont´ee de la temp´erature du combustible pourrait notamment en-traˆıner la fusion de la gaine m´etallique qui s´epare l’uranium de l’eau. Dans ce type de situation, il est donc imp´eratif d’injecter de l’eau rapidement, afin de tout “renoyer” pour faire baisser la temp´erature `a l’int´erieur de la cuve et ainsi

minimiser les cons´equences d’un APRP sur l’environnement.

Mod´

elisation et simulation num´

erique

La simulation num´erique des r´eacteurs nucl´eaires permet d’´etudier de telles configurations accidentelles, difficiles voire impossibles `a mettre en place en recherche exp´erimentale. Cependant, la complexit´e des ph´enom`enes physiques mis en jeu et la vocation industrielle de telles ´etudes entrainent diff´erents choix de mod´elisation d’´ecoulement diphasique eau - vapeur. Le choix de l’utilisation d’un mod`ele physique plutˆot qu’un autre se fait en g´en´eral en fonction de la pr´ecision dont on a besoin, et du coˆut en temps de calcul. Diff´erents mod`eles de thermohydraulique sont ainsi utilis´es dans diff´erentes parties du circuit. Dans l’industrie on parle de diff´erentes ´echelles de simulation :

• l’´echelle CFD ou 3D local n´ecessite une discr´etisation assez fine du do-maine de calcul pour permettre d’obtenir une simulation pr´ecise de l’´ecoulement dans une petite partie du r´eacteur nucl´eaire (une conduite, une pompe, par exemple)

• l’´echelle composant s’int´eresse `a ce qui se passe au niveau d’un des gros composants du circuit primaire (coeur du r´eacteur, g´en´erateur de vapeur,...) • l’´echelle syst`eme quant `a elle, permet d’obtenir une vision globale de

l’´ecoulement au sein du circuit primaire.

A cette classification se superpose celle li´ee aux mod`eles choisis. Les mod`eles utilis´es pour d´ecrire des ´ecoulements diphasiques eau - vapeur peuvent ˆetre class´es en deux cat´egories :

1. les mod`eles bifluides qui permettent en tout point de d´ecrire les car-act´eristiques physiques de l’eau et de la vapeur ind´ependamment, les deux phases ´etant repr´esent´ees par deux jeux d’´equations de bilan coupl´ees, 2. les mod`eles homog`enes qui consid`erent les caract´eristiques de l’´ecoulement

”global” (une seule vitesse, une seule temp´erature,...etc... pour les deux phases). Le m´elange diphasique est en quelque sorte assimil´ee `a un fluide moyen ´equivalent.

Actuellement, les codes de calcul bas´es sur des mod`eles bifluides et ceux bas´es sur des mod`eles homog`enes sont utilis´es dans des contextes diff´erents, en fonction d’une application donn´ee. On utilise g´en´eralement les mod`eles bifluides d’une part dans les codes syst`emes, d’autre part dans les codes CFD. Du fait de

leur capacit´e `a traiter toutes les situations `a l’´echelle composant, on utilise en g´en´eral les mod`eles homog`enes pour des raisons de compromis entre le temps de calcul et la pr´ecision n´ecessaire, sachant que le domaine de calcul est toujours le mˆeme.

Cependant, l’utilisation de codes de calcul bas´es sur des mod`eles homog`enes ne suffit pas toujours `a ´etudier certaines configurations industrielles `a l’´echelle composant dans des temps de calcul raisonnables. La g´eom´etrie du coeur du r´eacteur, par exemple, est extrˆemement complexe. Le combustible est empil´e sous forme de pastilles dans des tubes m´etalliques d’environ 4 m`etres de long appel´es ”crayons combustible”. Ces crayons sont regroup´es en assemblages. Un assemblage comprend un faisceau carr´e de 264 crayons (voir la photo 2), et un r´eacteur de 1300 MWe contient 193 assemblages 2 _{baignant dans l’eau sous}

pression contenue dans la cuve du circuit primaire. Pour pouvoir r´ealiser des calculs d’´ecoulements au sein du coeur du r´eacteur `a une ´echelle tr`es locale, il faudrait un maillage colossal ! C’est pour palier ce probl`eme que la formula-tion poreuse a ´et´e d´evelopp´ee dans le cadre des ´ecoulements diphasique eau - vapeur. Cette approche d’homog´en´eisation permet en effet de consid´erer que l’´ecoulement diphasique s’effectue dans un milieu poreux, ce qui permet de ne pas avoir `a mailler l’ensemble des petits obstacles qui composent le domaine d’´etude.

Figure 2: Photo d’un assemblage comprenant un faisceau carr´e de 264 crayons combustible (FRAMATOME ANP / E. Joly2)

Les codes industriels suivant l’approche poreuse utilisent principalement des mod`eles poreux homog`enes. Dans cette cat´egorie, les principaux codes fran¸cais sont:

• THYC 3_{, bas´e sur la m´ethode des volumes finis et d´edi´e aux coeur des}

r´eacteurs nucl´eaires et aux g´en´erateurs de vapeur, ce code est d´evelopp´e par EDF,

• FLICA4_{, bas´e sur la m´ethode des volumes finis, d´edi´e aux coeur des}

r´eacteurs nucl´eaires, et d´evelopp´e par le CEA,

• GENEPI5_{, bas´e sur la m´ethode des ´el´ements finis, d´edi´e aux g´en´erateurs}

de vapeur, et d´evelopp´e par le CEA.

Actuellement, le seul code bifluide et poreux est le code CATHARE6_{, dans}

son module 3D (les mod`eles bifluides ´etant habituellement utilis´es en milieu libre). D´evelopp´e par le CEA, ce code est lui aussi bas´e sur la m´ethode des volumes finis.

Le couplage interfacial de mod`

eles

La simulation du fonctionnement de syst`emes complexes n´ecessite parfois l’utilisation de plusieurs mod`eles afin de d´ecrire finement les comportements sp´ecifiques `a chaque composant comme nous l’avons vu plus haut. Des mod`eles physiques tr`es diff´erents les uns des autres sont utilis´es pour simuler les comportements fluides au sein des circuits hydrauliques d’une centrale nucl´eaire. Pour effectuer une simulation globale, on peut choisir de diviser le domaine d’´etude en sous-domaines s´epar´es par des interfaces fixes et extrˆemement minces appel´ees inter-faces de couplage. Des codes de calcul bas´es sur des mod`eles physiques diff´erents sont utilis´es sur ces sous-domaine. Le couplage interfacial permet alors de passer les informations d’un code de calcul `a un autre au travers de l’interface de cou-plage les s´eparant, dans l’optique d’obtenir une vision globale de l’´ecoulement `

a travers un circuit donn´e.

La contrainte principale du d´eveloppement de m´ethodes de couplage est de ne pas intervenir profond´ement dans les codes de calcul existants. Il faut mettre

3_{Aubry S., Cahouet J., Lequesne P., Nicolas G., Pastorini S., ”THYC: Code de}

Thermo-hydraulique des Coeurs de R´eacteurs. Version 1 .0 Mod´elisation et M´ethodes Num´eriques”, Rapport EDF H-T10-1988-02769-FR, 1988

4_{Toumi I., Bergeron A., Gallo D., Royer E., Caruge D., ”FLICA-4: a three-dimensional}

two-phase flow computer code with advanced numerical methods for nuclear applications”, Nuclear Engineering and design, vol. 200, pp. 139-155, 2000

5_{Grandotto M., ”Simulation num´erique des ´ecoulements diphasiques dans les ´echangeurs”,}

Habilitation `a Diriger des Recherches, Universit´e de Provence, 2006

6_{Serre G., Bestion D, ”Physical laws of CATHARE revision 6. Pipe module”, rapport}

CEA SMTH/LMDS/EM/98-038, 1999

en place des techniques permettant de passer l’information d’un code `a un autre sans pour autant les modifier, si ce n’est au niveau des conditions aux limites sur l’interface de couplage. Pour cela on d´ecide d’intervenir au niveau des con-ditions aux limites. Travaillant sur des mod`eles discr´etis´es par la m´ethode des volumes finis, les conditions de couplage seront donn´ees sous forme de flux. Ces flux de couplage, d´efinis au niveau des interfaces de couplage, devront prendre en compte les ´eventuelles incompatibilit´es entre les mod`eles `a coupler, bas´es sur des syst`emes d’´equations diff´erents. Le d´eveloppement de techniques de couplage consiste alors en grande partie `a d´efinir des conditions de couplage pertinentes afin d’obtenir une description coh´erente de l’´ecoulement vu dans sa globalit´e.

Fruit de la collaboration entre le laboratoire Jacques Louis Lions (JLL) de l’Universit´e Pierre et Marie Curie et le CEA (Commissariat `a l’Energie Atom-ique) de Saclay, un groupe de travail7a ´et´e cr´e´e `a Paris VI en 2004 afin d’´etudier les probl`emes de couplages de syst`emes d’´equations aux d´eriv´ees partielles non lin´eaires. Le but de ce groupe de travail est de d´evelopper des outils th´eoriques et num´eriques pour le couplage interfacial entre des mod`eles provenant de la m´ecanique de fluides multiphasiques. Plusieurs travaux de th`eses8, 9, 10, 11_ont

contribu´e `a la probl´ematique de couplage de mod`eles ´etudi´ee par le groupe de travail JLL-CEA et au d´epartement MFEE (M´ecanique des Fluides Energie et Environnement) d’EDF.

La principale m´ethode de couplage d´evelopp´ee par le groupe de travail JLL-CEA n´ecessite la r´esolution de deux probl`emes de Riemann au niveau de l’interface pour pouvoir d´efinir les flux de couplage. Une deuxi`eme approche peut ˆetre utilis´ee. Propos´ee pour le traitement des termes sources12_{, cette}

ap-proche n´ecessite d’introduire un ”mod`ele p`ere” au niveau de l’interface de cou-plage. Ce mod`ele p`ere doit ˆetre suffisamment riche pour pouvoir ”d´eg´en´erer” vers les mod`eles situ´es de part et d’autre de l’interface de couplage par une tech-nique de relaxation instantan´ee. Cette deuxi`eme approche sera utilis´ee dans la

7_{site internet du groupe de travail JLLCEA http://www.ann.jussieu.fr/groupes/cea/}

-Membres : Ambroso A., Chalons C., Coquel F., Godlewski E., Lagouti`ere F., Raviart P.A., Seguin N.

8_{Hurisse O., ”Couplage interfacial instationnaire de mod`eles diphasiques”, Th`ese de}

doc-torat de l’Universit´e Aix-Marseille I (2006)

9_{Caetano F., ”Sur certains probl`emes de lin´earisation et de couplage pour les syst`emes}

hyperboliques non lin´eaires”, Th`ese de doctorat de l’Universit´e Pierre et Marie Curie - Paris VI (2006)

10_{Gali´e T., ”Couplage interfacial de mod`eles en dynamique des fluides. Application aux}

´ecoulements diphasiques”, Th`ese de doctorat de l’Universit´e Pierre et Marie Curie - Paris VI (2009)

11_{Boutin B., ”Etude math´ematique et num´erique d’´equations hyperboliques non lin´eaires}

: couplage de mod`eles et chocs non classiques”, Th`ese de doctorat de l’Universit´e Pierre et Marie Curie - Paris VI (2009)

12_{Greenberg J.M., Leroux A.Y., ”A well balanced scheme for the numerical processing of}

derni`ere partie des travaux pr´esent´es ici.

Cette th`ese a ´et´e r´ealis´ee dans le cadre du projet de co-d´eveloppement NEP-TUNE, financ´e par le CEA, EDF, l’IRSN et AREVA NP. Dans ce projet sont d´evelopp´es en particulier le code NEPTUNE CFD 13 _{(code bifluide en milieu}

libre) et le code CATHARE 3 (code syst`eme - bifluide `a deux ou trois champs).

13_{Mechitoua N., Boucker M., Lavieville J., H´erard J.-M., Pigny S., Serre G., ”An}

unstruc-tured finite volume solver for two-phase water/vapour flows modelling based on an elliptic oriented fractional step method”, NURETH 10, 10thInternational Meeting on Nuclear Reac-tor, Thermal-hydraulics, Seoul, South Korea, October 5-9, 2003

Plan de la th`

ese

Dans cette th`ese nous souhaitons traiter des situations r´eellement instation-naires. Nous avons choisis d’utiliser des mod`eles en approche multifluide diphasique ou triphasique, en milieu libre ou poreux, o`u coexistent des ´echelles de temps de relaxation en vitesse, temp´erature, et pression, et bien pos´es `a conditions initiales.

Les probl`emes suivants sont abord´es :

• la mod´elisation bifluide en milieu poreux, lorsque le champ de porosit´e est r´egulier,

• le couplage entre milieu fluide et milieu poreux, lorsque le champ de porosit´e pr´esente de fortes variations locales,

• le couplage entre un mod`ele triphasique et un mod`ele diphasique.

Les deux premiers chapitres visent `a mettre en ´evidence une m´ethode de cou-plage fluide/poreux. L’objectif du premier chapitre est de pr´esenter diff´erents sch´emas num´eriques monodimensionnels pour la simulation d’´ecoulements dipha-siques dans des milieux poreux, puis d’´etudier leur comportement en cas de fortes variations de porosit´e. L’un de ces sch´emas donne clairement des r´esultats plus satisfaisants que les deux autres. On s’y int´eresse donc plus en d´etail dans le chapitre 2. Dans le deuxi`eme chapitre, on commence tout d’abord par ´etendre ce sch´ema au cas tridimensionnel avant d’´evaluer la pertinence des conditions de couplage au niveau de l’interface entre les milieux fluide et poreux.

Les travaux pr´esent´es dans ces deux premiers chapitres permettent d’exhiber une m´ethode de couplage entre milieu fluide et milieu poreux, pouvant ˆetre utilis´ee pour un couplage de codes de type NEPTUNE CFD — THYC, le code NEPTUNE CFD (mod`ele bifluide en milieu libre) ´etant alors utilis´e au niveau du circuit alors que THYC (mod`ele homog`ene en milieu poreux) est utilis´e au niveau du g´en´erateur de vapeur. Cette premi`ere partie permettrait aussi de d´efinir un code bifluide poreux g´erant divers niveaux de porosit´e.

Le troisi`eme et dernier chapitre vise quant-`a-lui `a proposer une m´ethode de couplage interfacial instationnaire entre un mod`ele triphasique et un mod`ele diphasique.

Chapitre 1: Comparaison de sch´emas pour la simulation d’´ecoulements diphasiques bifluides en milieu poreux

Chapitre 2: Evaluation de conditions de couplage au niveau de l’inter-face fluide/poreux et simulations multidimensionnelles

Chapitre 3: Proposition pour un couplage interfacial instationnaire d’un mod`ele triphasique et d’un mod`ele diphasique

2

Comparaison de sch´

emas pour la simulation

d’´

ecoulements diphasiques bifluides en milieu

poreux

Le d´eveloppement de codes diphasiques n´ecessite la construction de codes dits “composants” qui font appel ´a une formulation poreuse afin de r´ealiser des calculs r´ealistes de certaines configurations industrielles, sans avoir ´a mailler l’ensemble des petits obstacles qui les composent. Cette approche peut-ˆetre d´elin´ee sur la base de mod`eles diphasiques homog`enes ou d’approches ´a deux fluides. La construction des codes poreux existants se base sur une hypoth`ese de r´egularit´e de la distribution du champ spatial de porosit´e dans le domaine de cal-cul, et sur une approche poreuse homog`ene. Dans le cadre de la sˆuret´e nucl´eaire notamment, les ´etudes au sein des coeurs de r´eacteurs ou des g´en´erateurs de vapeur utilisent le concept de porosit´e dans des volumes de contrˆole. N´eanmoins, dans certaines configurations, cette hypoth`ese de r´egularit´e est clairement mise en d´efaut, car l’application vis´ee comporte des variations brusques de ce champ de porosit´e.

Le principal objectif de ce premier chapitre est de pr´esenter diff´erents sch´emas num´eriques pour la simulation d’´ecoulements diphasiques (eau liquide et vapeur d’eau) dans des milieux poreux, puis d’´etudier leur comportement dans le cas de fortes variations de porosit´e.

On commence donc par pr´esenter le mod`ele hyperbolique utilis´e ici pour simuler des ´ecoulements diphasiques dans un milieu poreux sur la base d’une approche bifluide, ainsi que ses propri´et´es. On d´efinit ensuite des solutions ana-lytiques simples et la construction de solutions exactes au probl`eme de Riemann unidimensionnel, qui serviront par la suite ´a calculer l’erreur en norme L1 pour les diff´erents sch´emas ´etudi´es.

Trois sch´emas aux Volumes Finis sont introduits dans ce chapitre: • Sch´ema de Rusanov classique (R)

• Sch´ema de Rusanov Modifi´e (M R)

• Sch´ema de type Well-Balanced Rusanov (W BR)

Ce dernier sch´ema est plus simple que le sch´ema well-balanced original pro-pos´e par Greenberg-Leroux qui n´ecessite la r´esolution exacte d’un probl`eme de Riemann ´a chaque interface. Le sch´ema W BR s’inspire du sch´ema propos´e par Kr¨oner et Thanh pour la simulation d’´ecoulements monophasiques dans des tuyaux ´a section variable. En effet, le sch´ema W BR ne requiert que la r´esolution de quatre ´equations scalaires non lin´eaires ´a chaque interface (deux pour chaque

phase), correspondant ´a la conservation des invariants de Riemann de l’onde sta-tionnaire associ´ee ´a la porosit´e. Les principales propri´et´es de ces sch´emas sont ensuite pr´esent´ees, comprenant notamment la pr´eservation de la positivit´e pour les taux de pr´esence et les masses partielles, sous condition CFL, et le respect ´eventuel des solutions stationnaires sur maillage de taille quelconque.

Diff´erents cas tests analytiques ont ´et´e r´ealis´es dans le but d’´evaluer, pour chacun de ces sch´emas, la norme L1 _{de l’erreur sur des maillages allant de}

100 mailles (maillage de taille industrielle) ´a 800 000 mailles. On s’est ensuite int´eress´e ´a des cas tests plus concrets, comme un accident de perte de r´efrig´erant primaire au sein d’une centrale nucl´eaire lors de la rupture d’une conduite (mi-lieu non poreux) menant au coeur du r´eacteur (consid´er´e comme un milieu poreux).

Les r´esultats fournis par le troisi´eme sch´ema W BR sont stables et conduisent ´

a des approximations convergeant vers la bonne solution pour des cas avec vari-ation brusque de porosit´e, ce qui n’est pas le cas des deux premiers sch´emas (R et M R).

Le respect par le sch´ema W BR des solutions stationnaires semble donc in-contournable pour obtenir une convergence correcte.

Le document qui suit a ´et´e accept´e pour publication ´a la revue M2_AN

(Mathematical Modelling and Numerical Analysis).

A two-fluid hyperbolic model in a porous medium

La¨

etitia Girault

EDF, R&D

Jean-Marc H´

erard

EDF, R&D

The paper is devoted to the computation of two-phase flows in a porous medium when applying the two-fluid approach. The basic formulation is presented first, together with the main properties of the model. A few basic analytic solutions are then provided, some of them corresponding to solutions of the one-dimensional Riemann problem. Three distinct Finite-Volume schemes are then introduced. The first two schemes, which rely on the Rusanov scheme, are shown to give wrong approximations in some cases involving sharp porous profiles. The third one, which is an extension of a scheme proposed by D. Kr¨oner and M. D. Thanh (27

) for the computation of single phase flows in varying cross section ducts , provides fair results in all situations. Properties of schemes and numerical results are presented. Analytic tests enable to compute the L1

norm of the error.

I.

Introduction

The main purpose of the present work is to develop models and schemes that allow the computation of two-phase flows in a porous medium while focusing on the two-fluid approach, and thus considering distinct pressure, velocity and temperature fields within each phase. Following the pioneering work of Baer and Nunziatto3_{, Kapila et al}25_{, and more recent work}8, 13_{, that deals with the two-fluid two-pressure approach}

in an open medium, we first provide an extension to the framework of porous medium as proposed in24 _.

As expected, the system is hyperbolic, at least for small enough Mach number within each phase, which of course seems reasonable when focusing on applications for flows in pressurised water reactors in nuclear power plants. The governing set of equations also benefits from two major features. Though it contains some non-conservative products, all jump conditions are unique in single genuinely non linear -GNL- fields. This is tightly connected with the fact that the closure law for the so-called interfacial velocity is such that the field associated with the eigenvalue λ = VI is linearly degenerated - LD- (see also8 ). Moreover, we emphasize

that a relevant entropy inequality holds for regular solutions of the whole set of equations including source terms (and viscous terms if any).

Once the model is presented (section II), section III will provide some details on a few analytic solutions of the whole system that will be investigated in sections V and V I. We will then focus in section IV on three simple Finite Volume schemes in order to compute approximations of the latter model in a porous medium. The first scheme corresponds to the classical Rusanov scheme, the second one being a slight modification of the latter. The third scheme is quite different. It relies on former propositions by Greenberg and Leroux (see20_{) revisited by Kr¨oner and Thanh (see}27_{, and}4_{too). Actually the latter third scheme does not require}

solving an exact Riemann problem around each cell interface (see19 _{), and thus is much simpler than the}

original well-balanced scheme20_{. The main properties of the schemes will be given in section V, with special}

emphasis on the well-balanced properties of course, but also on positivity properties. The last two sections will be devoted to the presentation of numerical results. More precisely, section V I will give the opportunity to examine the convergence rate of the above mentionned schemes with respect to the mesh size. Eventually, we will focus in section V II on the computation of the interaction of moving waves with the standing free/porous interface. Beyond the present work, we would like to mention that we aim at investigating some possible way to ensure a relevant interfacial and unsteady coupling of existing codes associated with free and porous medium respectively. This motivates the numerical experiments that are introduced within the last

∗_{PhD student, EDF, R&D, Fluid Dynamics, Power Generation and Environment, 6 quai Watier, 78400, Chatou, France.} Also in: Centre de Math´ematique et Informatique, LATP, 39 rue Joliot Curie, 13453, Marseille cedex 13, France

two sections. For a further insight on the interfacial coupling of models, we refer to the recent work by the working group0_{, and more precisely on the early work}17, 18_{, but also to the recent review article}16 _{, and}

also on5_.

II.

A two-fluid model in a porous medium

We first need to present the two-fluid two-pressure model introduced in24_{, and we shall also recall some}

of its properties afterwards.

A. Governing equations for the two-fluid model

We first introduce the void fraction αk ∈ [0, 1] that complies with α1+ α2 = 1, the porosity ǫ ∈]0, 1],

and (for k = 1, 2) the mean velocity Uk, the mean pressure Pk, the mean density ρk, the internal energy

ek= ek(Pk, ρk) in phase k. The state variable W in R8is:

Wt= (ǫ, α2, ǫm1, ǫm2, ǫm1U1, ǫm2U2, ǫE1, ǫE2) (1)

We will also use Wǫdefined as follows:

W_ǫt= (ǫmk, ǫmkUk, ǫEk) (2)

in R6_{, while noting m}

k= αkρk the partial mass in phase k, and Ek = mkUk2/2 + mkek the total energy of

phase k. The equation of state (EOS) is provided through the function ek(Pk, ρk), which may be arbitrary.

We will thus focus herein on the following two-fluid model:              ∂t(ǫ) = 0 ; ∂t(α2) + VI∂x(α2) = φ2(W ) ; ∂t(ǫmk) + ∂x(ǫmkUk) = 0 ; ∂t(ǫmkUk) + ∂x ǫmkUk2
+ ǫαk∂x(Pk) + ǫ(Pk− PI)∂x(αk) = ǫDk(W ) ; ∂t(ǫEk) + ∂x(ǫUk(Ek+ αkPk)) + ǫPI∂t(αk) = ǫψk+ ǫVIDk(W ) . (3)

We now detail the closure laws for the source terms (φ2, Dk, ψk), which agree with : 2 X k=1 ψk(W ) = 0 ; 2 X k=1 Dk(W ) = 0 ; 2 X k=1 φk(W ) = 0 . (4)

The latter two read:

(

Dk= (−1)k_(mm₁1_+mm2₂₎(U1− U2)/τU ;

φk= (−1)k_|Pα_1|+|P1α2_2|(P2− P1)/τP .

(5) where both τU and τP denote relaxation time scales. The contribution Dk refers to the drag forces. Besides,

the energy interfacial transfer term :

ψk= KT(ak− a3−k) (6)

requires to define ak:

ak= (sk)−1(∂Pk(sk))(∂Pk(ek))

−1 ₍₇₎

where sk= sk(Pk, ρk) denotes the specific entropy, which is compelled with:

(ck)2∂Pk(sk) + ∂ρk(sk) = 0 (8) noting as usual: (ck)2= (ck)2(Pk, ρk) = ( Pk (ρk)2 − ∂ρk(ek))(∂Pk(ek)) −1 ₍₉₎

The couple (VI, PI) is assumed to be one among the two couples (Uk, P3−k), with k ∈ 1, 2. For

in-stance, the pair (U2, P1) is expected to be physically relevant when the phase 2 is dilute (and reversely the

pair (U1, P2) when the flow is dominated by phase 2). these two pairs correspond to models investigated

in1, 3, 25, 28, 29 _{among others. Note anyway that a third choice corresponding to (V}

m, Pm) as defined in8, 13 is

also meaningful, and might be considered as well.

B. Main properties of the two-fluid model First, we focus on the homogeneous part of (3), eg:

             ∂t(ǫ) = 0 ; ∂t(α2) + VI∂x(α2) = 0 ; ∂t(ǫmk) + ∂x(ǫmkUk) = 0 ; ∂t(ǫmkUk) + ∂x ǫmkUk2
+ ǫαk∂x(Pk) + ǫ(Pk− PI)∂x(αk) = 0 ; ∂t(ǫEk) + ∂x(ǫUk(Ek+ αkPk)) + ǫPI∂t(αk) = 0 . (10)

Property 1 (Structure of the convective part of (3)):

The homogeneous system (10) admits the following real eigenvalues: λ0= 0 , λ1= VI,

λ2= U1 , λ3= U1− c1 , λ4= U1+ c1,

λ5= U2 , λ6= U2− c2 , λ7= U2+ c2

(11)

Associated right eigenvectors span the whole space if : |VI− Uk| 6= ck, and |Uk| 6= ck, for k = 1, 2.

Other-wise, a resonance phenomenum occurs. Fields associated with eigenvalues λ0, λ2, λ5are linearly degenerated

(LD), whereas fields associated with λ3, λ4, λ6, λ7are genuinely non-linear. Owing to the particular choice

VI= Uk, the field associated with λ1is also LD.

For nuclear applications with a mixture of water and vapour, resonance is very unlikely to happen, since material velocities are indeed small compared with the speed of acoustic waves in pure phases. For a more detailed investigation of resonance phenomena, we refer for instance to15 _{and also to}7 _{which focuses on}

shallow-water equations with topography. Property 2 (Entropy inequality):

Define the entropy-entropy flux pair (η, fη) as:

η = ǫ(m1Log(s1) + m2Log(s2))

fη= ǫ(m1Log(s1)U1+ m2Log(s2)U2)

and the following quantity:

mkRk= ak(ψk+ Dk(VI− Uk) − φk(PI− Pk)) (12)

Then smooth solutions of system (3) comply with the following entropy inequality :

∂t(η) + ∂x(fη) = ǫ(m1R1+ m2R2) ≥ 0 . (13)

Before going further on, it may be noticed that slightly different choices of PI might be considered

(see24, 8, 13_{), which result in a dissipative contribution in the governing equation of the entropy η. We insist}

that these are not considered in the present paper. Property 3 (Riemann invariants in the standing wave):

The linearly degenerated (LD) wave associated with λ = 0 admits the following Riemann invariants I0 1(W ) = α2 ; I20(W ) = s1 ; I30(W ) = ǫm1U1 ; I0 4(W ) = e1+P_ρ11+U 2 1 2 ; I50(W ) = s2 ; I0 6(W ) = ǫm2U2 ; I70(W ) = e2+P_ρ22+U 2 2 2

Property 4 (Riemann invariants in the VI contact discontinuity):

We still assume that VI = Uk. As a consequence, the wave associated with the eigenvalue λ = VI is linearly

degenerated. Moreover, associated Riemann invariants are the following: I1 1(W ) = ǫ ; I21(W ) = s3−k ; I1 3(W ) = Uk ; I41(W ) = m3−k(U3−k− Uk) ; I1 5(W ) = α1P1+ α2P2+ m3−k(U3−k− Uk)2 ; I1 6(W ) = e3−k+P_ρ3−k_3−k +1₂(U3−k− Uk)2

The latter property is obviously extremely important. Actually, even in a free medium, thus corresponding to the uniform distribution ǫ = 1, the specific closure law for the so-called interfacial velocity-pressure pair (VI, PI) guarantees that the non-conservative products are only active in a linearly degenerated field.

Thus unique jump conditions hold field by field, which results in the crucial point that the converged approximations (w.r.t. the mesh size) obtained when computing flows with shock waves through system (3) will not depend on the scheme (see21_{), as may happen for other unsuitable choices of (V}

I, PI).

III.

Basic solutions

A. Two simple solutions

We define two basic solutions of system (3), whatever the EOS is. • Basic solution S1:

We define solution S1as the following unsteady solution:

     ǫ(x) = ǫ0 P1(x, t) = P2(x, t) = P0 U1(x, t) = U2(x, t) = U0 (14)

while both ρk and α2are solutions of the governing equation:

∂t(f ) + U0∂x(f ) = 0

Note that this solution, which is only valid in a free medium, may be viewed as a solution of the sole convective part of system (3), or alternatively of the full set of equations (3).

• Basic solution S2:

We assume that the distribution ǫ(x) is arbitrary. Solution S2will correspond to the steady solution:

(

P1(x, t) = P2(x, t) = P0

U1(x, t) = U2(x, t) = 0

(15)

while both mk(x, t) = mk(x, 0) and α2(x, t) = α2(x, 0).

These two basic solutions will be used in order to define a priori suitable schemes. The second solution S2will also be used as a preliminary test case (test 2) in numerical experiments in section VI.

B. Solutions of the Riemann problem

We focus here the homogeneous part of system (3), and thus consider now solutions of the Riemann problem associated with system (10). We must insist here that we do not know whether there exists a unique solution to the one dimensional Riemann problem for our problem, given left and right initial states. However, we may proceed differently and construct exact solutions. For that purpose we simply introduce a left initial condi-tion, and then construct intermediate states and associated single waves, choosing an initial configuration. We shall restrict here to rather simple choices involving ”ghost” waves -through which no variation of the state variable occurs-, and: (i) a steady contact discontinuity, and/or (ii) a moving VI contact discontinuity

and/or (iii) a shock wave in phase 2. Solutions will be referred to as Test 1, Test 3 and Test 4 respectively.

Of course much more complex test cases might be defined that way, such as those examined in1, 28, 29 _for

instance, but we emphasize here once more that we want to focus on situations where the porosity varies, which explains our choices. The exact construction of intermediate states and the final right state is detailed in appendix A. Numerical values that are used in numerical experiments are recalled at the beginning of section VI, which is devoted to the measure of the L1_{norm of the error.}

The figure below provides a sketch of the solution of Riemann problems that will be investigated, together with notations :

Figure 1. Sketch of the specific fan of waves in exact solutions and notations for intermediate states.

IV.

Finite Volume schemes

We introduce standard notations for Finite Volume schemes (see9_{). Within each Finite Volume of size}

hi= xi+1/2− xi−1/2, the mean value of W at time tnin cell i is:

W_in= ( Z

[xi−1/2,xi+1/2]

W (x, tn)dx)/hi (16)

The time step ∆tn _{will comply with a standard CFL condition. Moreover, we define:}

ai+1/2= (ai+ ai+1)/2

whatever the quantity a is, and also:

(∆(a))ni = (a)ni+1/2− (a)ni−1/2

for k = 1, 2.

We define the flux fǫin R6:

fǫ(Wǫ, α2, ǫ)t= (ǫmkUk, ǫmkUk2, ǫUk(Ek+ αkPk)) (17)

The computation of the whole set (3) is achieved with a fractional step method which is in agreement with the overall entropy inequality. The homogeneous problem associated with (10) is computed first. Source terms are then accounted for using an implicit scheme, which is exactly the one described in reference13 _.

We thus only describe the first evolution step here. A. Classical Rusanov scheme R

The cell scheme which is used to compute the evolution step simply reads:

where cn

i+1/2= −rni+1/2((α2)ni+1− (α2)ni)/2, while:

hi((Wǫ)n+1i − (Wǫ)ni) + ∆tn(Fi+1/2R ((Wǫ)nl, (α2)ln, ǫl) − Fi−1/2R ((Wǫ)nl, (α2)nl, ǫl)) + ∆tn(Hǫ)ni = 0 (19)

where the numerical flux is defined by: FR

i+1/2((Wǫ)nl, (α2)nl, ǫl) =



fǫ((Wǫ)ni, (α2)ni, ǫi) + fǫ((Wǫ)ni+1, (α2)ni+1, ǫi+1) − rni+1/2((Wǫ)ni+1− (Wǫ)ni)

 /2 (20) The notation (Wǫ)nl involves the stencil l that refers to cell indices (i, i + 1) for Fi+1/2, (respectively to

(i − 1, i) for Fi−1/2). The scalar rni+1/2represents the maximal value of the spectral radius of the Jacobian

matrices A((Wǫ)nl, (α2)nl, ǫl) for l = i, i + 1. The contribution connected with the first-order non-conservative

terms (Hǫ)ni is approximated by:

(Hǫ)ni = (0, ǫi(((Pk)ni − (PI)ni)(∆(αk))ni + (αk)ni(∆(Pk))ni), −ǫi(PI)in(VI)ni(∆(αk))ni) (21)

B. A modified Rusanov scheme M R

This scheme is similar to the previous one. Its main interest is that it guarantees that the steady solution S2will be perfectly approximated on any mesh size (see section V below).

The update for the void fractions is:

hi((α2)n+1i − (α2)ni) + ∆tn(VI)ni(∆(α2))ni + ∆tn(dni+1/2,−− dni−1/2,+) = 0 (22)

where :

dn

i+1/2,−= −(ˆǫ)i+1/2rni+1/2((α2)ni+1− (α2)ni)/(2ǫi)

dn

i−1/2,+= −(ˆǫ)i−1/2rni−1/2((α2)ni − (α2)ni−1)/(2ǫi)

(23)

The numerical flux in (19) is replaced by:

FM R

i+1/2((Wǫ)nl, (α2)nl, ǫl) =



fǫ((Wǫ)ni, (α2)ni, ǫi) + fǫ((Wǫ)ni+1, (α2)ni+1, ǫi+1) − rni+1/2(ˆǫ)i+1/2((Wǫ)

n i+1 ǫi+1 − (Wǫ)ni ǫi )  /2 (24) where (ˆǫ)i+1/2= max(ǫi, ǫi+1), or : (ˆǫ)i+1/2= (2ǫiǫi+1)/(ǫi+ ǫi+1).

C. A simplified well-balanced scheme W BR

The basic idea is the following. For the sake of simplicity, we introduce Z ∈ R7_{and f (Z) ∈ R}7_{as follows:}

Zt_{= (α}

2, mk, mkUk, Ek)

f (Z)t_{= (0, m}

kUk, mkUk2+ αkPk, Uk(Ek+ αkPk))

(25)

Now, since ǫ is assumed to be constant within each cell, the cell scheme will read:

hi(Zin+1− Zin) + ∆tn(Fi+1/2,−W BR (Zln, ǫl) − Fi−1/2,+W BR (Zln, ǫl)) + ∆tnHin= 0 (26)

where the numerical fluxes and the contribution H are defined by: FW BR

i+1/2,−(Zln, ǫl) =

 f (Zn

i) + f (Zi+1/2,−n ) − (rW B)ni+1/2(Zi+1/2,−n − Zin)

 /2 FW BR i−1/2,+(Zln, ǫl) =  f (Zn

i) + f (Zi−1/2,+n ) − (rW B)ni−1/2(Zin− Zi−1/2,+n )



/2 (27) Hn

i = ((VI)ni(∆(α2))ni, 0, −(PI)ni(∆(αk))ni, −(PI)ni(VI)ni(∆(αk))ni) (28)

The values Zn

i−1/2,+and Zi+1/2,−n are obtained by solving the non-linear equations (for m = 0 to 6):

Inv0

m(Zi−1/2,+n , ǫi) = Invm0(Zi−1n , ǫi−1)

Inv0

m(Zi+1/2,−n , ǫi) = Invm0(Zi+1n , ǫi+1)

(29)

In agreement with section II (property 3), we have set here: Inv0 0(Z, ǫ) = α2 Inv0 3k−2(Z, ǫ) = sk Inv0 3k−1(Z, ǫ) = ǫmkUk Inv0 3k(Z, ǫ) = ek+ Pk/ρk+ Uk2/2 (30)

for k = 1, 2. In practice, this requires solving two uncoupled non-linear scalar equations (one for each phase) at each cell interface i + 1/2, the solution of which is trivial when ǫi = ǫi+1, or when (Uk)ni˙(Uk)ni+1 = 0.

Details pertaining to the exact solution of the above-mentionned equations are given in appendix C. We emphasize here that:

max(|(Uk)ni+1/2,−|, |(Uk)i+1/2,+n |, rnl) = (rW B)ni+1/2 for k = 1, 2

where rn

l stands for the spectral radius of the Jacobian matrix at time tn for l = i, i + 1.

We note that the update for α2 is exactly the same as the one achieved in (18), owing to the specific

value of Inv0

0(Z, ǫ) (which implies that : (α2)ni+1/2,−= (α2)ni+1and (α2)ni−1/2,+= (α2)ni−1). Obviously when

the porosity is uniform (ǫi−1 = ǫi = ǫi+1), this scheme identifies with the standard Rusanov scheme, since

Zn

i−1/2,+= Zi−1n and Zi+1/2,−n = Zi+1n in that case, and it also corresponds to scheme M R.

V.

Main properties

We wonder first whether the latter three schemes preserve basic solutions on any mesh, which is of course crucial for industrial applications. For that purpose, we need to introduce some constants ak,0 for both

phases, together with two invertible functions gk(φ). Actually, one may easily check that:

• Property 5: We assume that the EOS takes the form: ρkek(Pk, ρk) = ak,0ρk+ gk(Pk) in each phase

k. The three schemes R, M R and W BR described above preserve the discrete form of the basic so-lution S1, whatever the mesh size is, since (U1)ni = (U2)ni = U0 and (P1)ni = (P2)ni = P0 imply that

(U1)n+1i = (U2)n+1i = U0 and (P1)n+1i = (P2)n+1i = P0, if ǫi= ǫ0.

The reader is referred to appendix B, section A for proof, which is almost obvious. In practice, stan-dard EOS such as perfect gas EOS or stiffened gas EOS belong to the above mentionned class. We recall that the stiffened gas EOS simply stands for ak,0 = 0 and gk(φ) = (φ + γkΠk)/(γk− 1), where

constants γk and Πk are assumed to be such that: γk> 1 and 0 ≤ Πk.

Of course (see10 _{for such a discussion in the framework of homogeneous models), it does not mean a}

priori that the schemes will - or won’t- converge towards correct solutions.

The next property is also useful for practical applications, though it is not sufficient of course. It requires similar assumptions on the form of the EOS.

• Property 6: We assume that the EOS takes the form: ρkek(Pk, ρk) = ak,0ρk+ gk(Pk) in each phase

k. Both schemes M R and W BR preserve the discrete form of the basic solution S2 on any mesh,

since (U1)ni = (U2)ni = 0 and (P1)ni = (P2)ni = P0 imply that (U1)n+1i = (U2)n+1i = 0 and also

(P1)n+1i = (P2)n+1i = P0, with arbitrary ǫi. The standard R scheme does not.

A proof is detailed in appendix B, section B, which is again almost obvious, and only requires a few calculations. The structure of the scheme M R with respect to the void fraction is of course mandatory to ensure the result.

• Property 7: We use notations introduced in section II pertaining to Riemann invariants. If we assume that ǫL 6= ǫR, and also that the initial conditions (WL, WR) of the Riemann problem comply with :

I0

m(WL) = Im0(WR), for m = 1 to 7, we are ensured that the scheme W BR preserves steady states on

any mesh. This does not hold true for schemes R and M R.

The proof for schemes R and M R is not detailed here since it is obvious. The one pertaining to scheme W BR is given in appendix B, section C. Actually, it is also close to some results stated in4_{. The proof}

is also almost the same as the one given in26_{in the case of Euler equations with perfect gas EOS, in a}

one-dimensional framework, where authors examine the particular case of flows in variable cross section ducts. It occurs in fact in the proof that, though the present system is indeed much more complex than the one examined in26_{, both phases almost ”decouple” through the interface, since the void fraction}

is one among the seven Riemann invariants of the standing wave associated with λ0(see section II). A

straightforward consequence is that the governing equations for the mass, momentum and total energy within phase k in a porous medium almost behave ”locally” as the Euler equations in a porous medium.

• Property 8: The maximum principle for the void fractions holds, and positive cell values of partial masses are ensured when applying any scheme among R, M R and W BR, provided that the following CFL conditions hold:

R scheme:

∆tn

2hi

(r_i+1/2n + rn_i−1/2) ≤ 1 ∀i (31)

M R scheme (with (ˆǫ)i+1/2= max(ǫi, ǫi+1)):

∆tn 2hi _(ˆǫ) i+1/2 ǫi r_i+1/2n +(ˆǫ)i−1/2 ǫi rn_i−1/2  ≤ 1 ∀i (32) W BR scheme: ∆tn 2hi

((rW B)ni+1/2+ (rW B)ni−1/2) ≤ 1 ∀i (33)

Proofs are given in appendix B, section D.

These results were expected, owing to the specific structure of the underlying Rusanov scheme. The CFL-like condition is almost classical for both R and W BR schemes, and it is slightly different for the M R scheme.

VI.

Convergence rate for analytic solutions

We examine in this section the true convergence rate of the above mentionned schemes, when computing Riemann problems as explained in section III. We do not present all results for the three schemes in any case, but we concentrate on the main features, drawbacks and advantages of schemes.

For all cases, we use uniform meshes, and the range of the mesh size will be recalled in each case. The coarser mesh contains 100 cells, whereas the finer mesh contains 8.105_{cells. More precisely, we use:}

Schemes R MR WBR

Test case 1 102_{to 2.10}5_cells

Test case 2 102_{to 2.10}5_{cells 10}2_{to 2.10}5_{cells 10}2_{to 2.10}5_cells

Test case 3 102_{to 4.10}5_{cells -}

-Test case 4 102_{to 4.10}5_{cells 10}2_{to 4.10}5_{cells 10}2_{to 8.10}5_cells

The EOS for the vapour phase (k = 2) and the water phase (k = 1) are assumed to be perfect gas EOS, and the corresponding constants will be γ1= 1.1 and γ2= 1.4.

The measure of the L1 _{norm of the error will be provided, together with a ”local” estimation of the}

convergence rate when meaningful. To be more complete, we recall that we use the pair (VI, PI) = (U2, P1).

In all computations, we use a CFL number 1/2 in order to compute the value ∆tn _{at each time step. We}

also recall below the main configurations that will be investigated, as explained in section III.

TEST 1 : Free medium TEST 2 : Solution S2

TEST 3 : double contact discontinuity TEST 4 : three-wave pattern

A. Test case 1: A two-wave Riemann problem in a free medium

The first solution corresponds to a very simple flow pattern in a free medium. It only involves one void frac-tion contact-discontinuity (associated with λ = U2), and a shock wave in the vapour phase corresponding to

the eigenvalue λ7= U2+ c2. Thus left and right initial states are separated by an intermediate state labelled

B. We provide below the exact initial data. We recall that results obtained with R, M R, W BR schemes are identical for this test case in a free medium.

state L state B state R

ǫ 1 α1 0.95 0.05 ρ1 1 0.956131034 U1 10 −84.3587663 P1 100000 95185.1407 ρ2 0.1 0.15 0.1 U2 15 −357.299567 P2 10000 95044.7777 53462.6875

Computations have been performed using regular meshes with 102_{, 5.10}2_{, 10}3_{, 5.10}3_{, 10}4_{, 5.10}4_{, 10}5_{, and}

2.105_{cells. These enable to plot the L}1_{norm of the error e}

h-in logarithmic coordinates- in Figure 2.

One may deduce the measure of the rate of convergence β at time t = T , while focusing on the four finer meshes, and enforcing the behaviour eφ_h(T ) = Cφ(T )hβ(φ).

0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 U₁ P₂ α₁ ρ₂ P 1 ρ₁ U₂ Ch1/2

Figure 2. L1 norm of the error for the R scheme when computing test case 1 (flow in a free medium) as a function of the mesh size.

between 104_{and 5.10}4_{cells between 5.10}4_{and 10}5_{cells between 10}5_{and 2.10}5_cells

α1 0.500 0.500 0.500 ρ1 0.494 0.496 0.497 U1 0.499 0.500 0.500 P1 0.497 0.498 0.498 ρ2 0.509 0.505 0.503 U2 0.919 0.846 0.797 P2 0.505 0.503 0.502

These values of β(φ) were actually expected. When restricting to ρ1, U1, P1, and on α1which only vary

in this test case through the void fraction contact discontinuity, an asymptotic rate of 0.5 is ”perfect”. Moreover, since (ρ2, P2) vary through both waves, the same is expected. A contrario, the rate β(U2) should

be close to 1 since U2is a Riemann invariant through the void fraction contact discontinuity. As a matter

of fact, the measured value seems to be close to 0.8, and this agrees with measurements performed in11 _.

B. Solution S2: a simple steady flow with a free/porous interface

We now focus on the behaviour of schemes R, M R and W BR when computing approximations of solution labelled S2, whose initial data is given below:

state L state R ǫ 1 0.6 α1 0.95 0.05 ρ1 1 U1 0 P1 100000 ρ2 2 0.15 U2 0 P2 100000 10 of 27

Results with R scheme

Simulations involve meshes with 102_{, 5.10}2_{, 10}3_{, 5.10}3_{, 10}4_{, 5.10}4_{, 10}5_{, and 2.10}5 _{regular cells. Figure 3}

displays the L1_{norm of the error e} h. 0.001 0.01 0.1 1 0.001 0.01 0.1 1 10 U₂ U₁ ρ₂ α₁ P₂ P₁ ρ₁ Ch1/2

Figure 3. Test 2: L1norm of the error for scheme R.

The R scheme clearly no longer converges towards the correct solution when the mesh size goes to zero. The illusion that α1and ρ2still converge is due to the fact that their initial condition is discontinuous, which

dissimulates the uncorrect behaviour on these ”coarse” meshes (wrt to what is seeked !). This renders the whole rather dangerous: if we restrict to a range of meshes with 100 up to 10000 cells (the latter represents a ”fine mesh” for practical applications...), the R scheme looks almost correct, and one may expect that the pollution will reduce when h tends towards 0...which is obviously not true. Actually, estimations of ”convergence rates” β(φ) on the finer meshes are 0.067, 0.017, 0.014 for ρ1, U1and P1respectively.

Results with M R scheme

The meshes are exactly the same. The L1_{norm of the error e}

hhas been plotted in figure 4. Restricting to

the finer meshes, approximations of convergence rates β(φ) for ρ2and α1are clearly 1/2. Round-off errors

are observed for all other variables. .

Results with W BR scheme

Meshes are still the same, and results are very similar to those obtained with the latter scheme M R. The L1

norm of the error ehhas been plotted in figure 5. We have also plotted in figures 6 approximations of both

densities ρ1, ρ2 that have been obtained using R, M R, W BR schemes on a rather coarse mesh with 1000

nodes. This clearly shows the poor accuracy of the Rusanov scheme: the approximations are spurious around the steady interface, and fast waves propagate on both sides apart from the coupling interface x = 0.5. This is of course even more astonishing when looking at the density profile in phase 1.

C. Test 3: a combination of a standing contact wave and a void fraction contact discontinuity This test case is similar to the second one, but it also involves a contact discontinuity associated with the void fraction wave (λ5= U2). The intermediate state A has been calculated using appendix A. Meshes now

0.001 0.01 0.1 1 1e-16 1e-12 1e-08 0.0001 1 ρ₂ α₁ U 2 U₁ P₂ ρ₁ P₁ Ch1/2

Figure 4. Test 2: L1 norm of the error for scheme M R.

0.001 0.01 0.1 1 1e-16 1e-12 1e-08 0.0001 1 ρ₂ α₁ Ch1/2 U₁ U₂ P 2 ρ₁ P₁

Figure 5. Test 2: L1norm of the error for scheme W BR.

state L state A state R

ǫ 1 0.6 α1 0.95 0.05 ρ1 1 0.999190167 0.853058301 U1 10 16.6801748 −160.919041 P1 100000 99910.922 83960.8032 ρ2 0.1 0.0998565629 0.15 U2 15 25.0359108 P2 10000 9979.92457 94534.4211 12 of 27

0 0.2 0.4 0.6 0.8 1 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2

Figure 6. Test 2: density profiles ρ1, ρ2 when using schemes R -blue line-, M R -red line with circles-, W BR -black dotted line- on a 1000-cell mesh.

The L1_{norm of the error is plotted in figure 7 when focusing on scheme R.}

0.0001 0.001 0.01 0.1 1 0.001 0.01 0.1 1 U₂ P 2 U₁ ρ₂ α₁ P 1 ρ₁ Ch1/2

Figure 7. Test 3: L1norm of the error for scheme R.

Once again, we may check that the R scheme no longer converges towards the correct solution, which is of course in agreement with the results of test case 2. We emphasize again that the behaviour on the coarser meshes (on the right side in figure 7) is somewhat misleading.

D. Test4: a three-wave pattern

This solution contains two contact waves associated with λ0= 0 and λ1= λ5= U2, and one shock wave in

state L state A state B state R ǫ 1 0.6 α1 0.95 0.05 ρ1 1 0.999190167 0.853058301 U1 10 16.6801748 −160.919041 P1 100000 99910.922 83960.8032 ρ2 0.1 0.0998565629 0.15 0.1 U2 15 25.0359108 −346.262753 P2 10000 9979.92457 94534.4211 53175.6119

Still using meshes with 102_{, 5.10}2_{, 10}3_{, 5.10}3_{, 10}4_{, 5.10}4_{, 10}5_{, 2.10}5_{and 4.10}5_{cells, we check first that the}

standard Rusanov scheme does not converge towards the correct solution, and then turn to the M R scheme. Results with M R scheme

First we provide in figure 8 results obtained for α2ρ2when computing the test case on a regular mesh with

one thousand cells, together with the exact solution. We still consider meshes used in preceeding tests, and errors computed for the M R scheme are given in Figure 9 -on the left side-. The crucial point that occurs is that estimations of convergence rates show evidence that the M R scheme no longer converges towards the correct solution. This is quite obvious when focusing on the U2profile in Figure 9. An approximation of the

convergence rate for U2on the finer meshes is 0.035. It obviously means that the ”static” property 6 is far

from being sufficient to guarantee convergence of approximations towards the correct solution.

0 0,2 0,4 0,6 0,8 1 0 0,02 0,04 0,06 0,08

Figure 8. Test 4: mass fraction α2ρ2 when using M R scheme -black line with crosses- on a 1000-cell mesh, together with the exact solution - red dashed line-.

Results with W BR scheme

Using exactly the same meshes and a finer mesh with 8.105_{cells, it occurs in Figure 9 (on the right side)that}

convergence towards the correct solution is now recovered with W BR. Moreover, we retrieve expected rates of convergence β(φ) that are close to 1/2, since all components φ vary through at least one contact discontinuity in this fourth test case.

0.001 0.01 0.1 1 0.001 0.01 0.1 U2 P2 U1 _α 1 ρ₂ P_ρ1 1 Ch1/2 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 U1 P2 α1 ρ₂ P1 ρ₁ U2 Ch1/2

Figure 9. Test 4: L1norm of the error for scheme M R (left) and W BR (right) schemes.

5.104_{to 10}5_{cells 10}5_{to 2.10}5_{cells 2.10}5_{to 4.10}5_{cells 4.10}5_{to 8.10}5_cells

α1 0.535 0.521 0.512 0.506 ρ1 0.495 0.493 0.495 0.496 U1 0.499 0.495 0.497 0.497 P1 0.496 0.494 0.496 0.496 ρ2 0.607 0.562 0.529 0.516 U2 0.625 0.655 0.505 0.521 P2 0.607 0.560 0.529 0.516

Local behaviour around the steady interface

We show in figure 10 the behaviour of the discharge ǫm1U1 around the steady interface x = 0.5 when

computing the Riemann problem discussed above with M R and W BR schemes, while restricting to a coarse and a rather fine mesh. The W BR scheme provides a reasonable local behaviour around the steady interface, and the amplitude of the reflected wave is indeed much smaller than the one occuring with the M R scheme. Similar remarks hold for ǫm2U2and other 0-Riemann invariants.

0 0.2 0.4 0.6 0.8 1 -5 0 5 10 15 20 Q1 0.46 0.48 0.5 0.52 2 4 6 8 10 12

Figure 10. Mean momentum ǫm1U1 obtained with M R and W BR -dotted line- schemes. The coarse and fine meshes respectively contain 500 -in black- and 50000 -in red- cells. A zoom on [0.46, 0.54] on the right figure provides some more details.

VII.

Interaction of fluid waves with the steady interface

We now examine two tests where waves issuing from the free medium propagate towards the porous interface and interact with it. Though we have no analytic solution available in that case, it represents some relevant pattern for industrial applications. The validation of approximations provided by schemes will be evaluated by focusing on the Riemann invariants of the steady wave computed within each cell, when the flow is almost steady around the interface. Once again, the CFL number is equal to 1/2 in all cases. Fifth test case

This test case corresponds to a rough representation of a loss of coolant accident, where we focus on the propagation of the rarefaction wave that will hit a free/porous interface separating the pipe from the steam generator. The computational domain includes a free region (ǫ(x < 0.35) = 1) on the left side of an interface located at x = 0.35, and a porous region (ǫ(x > 0.35) = 0.6) on the right side of the latter interface. The whole computational domain thus corresponds to x ∈ [0; 1]. The pipe is suddenly broken around x = 0.30, at the beginning of the computation (t = 0). Denoting by L, R the left and right states on both sides of the interface x = 0.30, the initial conditions are:

Left state L Right state R

ǫ 1 0.6 α1 0.95 0.05 ρ1 1 U1 0 P1 1.105 1.106 ρ2 20 U2 0 P2 1.105 1.106

Figures 11 and 12 show the behaviour of the Riemann invariants H1, H2, s1and s2, when the rarefaction

has passed the free/porous interface. The mesh contains 1000 regular cells. Subscript 2 now refers to the water phase. The best results are once again obtained with the W BR scheme. Those pertaining to the M R scheme are less spurious than those corresponding to the Rusanov scheme, but show again a non-monotone behaviour. 0 0.2 0.4 0.6 0.8 1 0 1e+06 2e+06 3e+06 4e+06 H₁ H₂ 0.3 0.35 0.4 2.85e+06 2.9e+06 2.95e+06 3e+06 3.05e+06 3.1e+06 3.15e+06

Figure 11. Test case 5: Riemann invariants H1and H2, using a mesh with 1000 cells. The blue line, the red line with circles and the dotted black line correspond to R, M R and W BR schemes respectively. A zoom around the steady interface is displayed on the right for H1.

0 0.2 0.4 0.6 0.8 1 0 5e+05 1e+06 1.5e+06 2e+06 s₁ s₂ 0.25 0.3 0.35 0.4 8.5e+05 9e+05 9.5e+05 1e+06 1.05e+06 1.1e+06

Figure 12. Test case 5: Riemann invariants s1and s2, using a mesh with 1000 cells. The blue line, the red line with circles and the dotted black line correspond to R, M R and W BR schemes respectively. A zoom around the steady interface is displayed on the right for s1.

Sixth test case

The initial conditions for this last test case are given in the table below. The computational domain cor-responds to x ∈ [0; 1]. The interface between codes is situated at x = 0.67, and the porous medium is still on the right side of this interface. Both phases are at rest at the beginning of the computation, and L, R states denote the initial states on the left and right side of x = 0.65 respectively. Results for H1, H2, s1and

s2 are displayed in figures 13 and 14. These have been plotted after the right-going shock waves and the

right-going void fraction contact discontinuity have passed the coupling interface (x = 0.67).

Left state L Right state R

ǫ 1 0.6 α1 0.05 0.95 ρ1 1 U1 0 P1 1.106 1.105 ρ2 20 U2 0 P2 1.106 1.105

This last test case confirms that the behaviour of W BR scheme is indeed fair.

VIII.

Conclusion

• None among the first two schemes R and M R converges towards the correct solution when refining the mesh, if one computes approximations of solutions in very simple one-dimensional Riemann problems involving distinct values of porosity, such as those introduced in section III. Actually, the continuity that is enforced by the Rusanov scheme seems to inhibit the correct convergence when focusing on models for porous media. This is not due to the complexity inherent to the two-fluid approach, and the poor behaviour of these schemes in the framework of Euler equations in variable cross section flows has already been pointed out in the literature (see27 _{for instance). However, and to the knowledge of}

authors, it had never been clearly stated that this drawback was not only annoying for coarse meshes, but also when tackling very fine meshes.

• On the contrary, the W BR scheme, which inherits the spirit of well-balanced schemes combined with the inner stability of Rusanov-like fluxes, provides an extremely simple and useful tool for such com-putations in porous media. The expected rates of convergence are retrieved, focusing on so-called

0 0.2 0.4 0.6 0.8 1 0 1e+06 2e+06 3e+06 4e+06 H₁ H₂ 0.5 0.6 0.7 0.8 0.9 2.5e+06 3e+06 3.5e+06

Figure 13. Test case 6: Riemann invariants H1and H2, using a mesh with 1000 cells. The blue line, the red line with circles and the dotted black line correspond to R, M R and W BR schemes respectively. A zoom around the steady interface is displayed on the right for H1.

0 0.2 0.4 0.6 0.8 1 0 5e+05 1e+06 1.5e+06 2e+06 s₁ s₂ 0.55 0.6 0.65 0.7 0.75 0.8 0.85 6e+05 7e+05 8e+05 9e+05 1e+06 1.1e+06 1.2e+06

Figure 14. Test case 6: Riemann invariants s1and s2, using a mesh with 1000 cells. The blue line, the red line with circles and the dotted black line correspond to R, M R and W BR schemes respectively. A zoom around the steady interface is displayed on the right for s1.

first-order schemes, and the nice behaviour of the discrete cell-values of Riemann invariants of the standing wave around the free/porous interface renders the scheme quite appealing. We recall that this scheme is inspired by the scheme introduced in27_{for Euler equations in variable cross section ducts}

(see26 _{too). It also means that the property 6 is far from being sufficient, and that the dynamical}

well-balanced property 7 enjoyed by W BR, the continous counterpart of which is property 3, seems mandatory to obtain convergence towards the correct solution.

A straightforward consequence for the NEPTUNE project is that at least one meaningful scheme is avail-able in order to perform the interfacial unsteady coupling of codes that aim at providing approximations of PDEs in free and porous medium respectively.

Among possible improvements and current work in progress in this area, we would like to mention that: • In order to achieve a better coupling, and more precisely a more relevant treatment of the free/porous interface, we need to provide more physical coupling conditions for momentum equations that might account for the head loss of momentum. This is known to be very tricky, and it is not clear whether direct simulations will in fine provide useful tools in practice. A similar remark holds for heat losses through the free/porous interface.

• We also would like to emphasize that an approximate Godunov scheme that inherits a similar well-balanced property may be constructed. This one also makes use of basic ideas introduced by Greenberg and Leroux (20_{), while substituting the approximate Godunov VFRoe-ncv fluxes introduced in}6_{to the}

exact Godunov ”fluxes” through the cell interfaces, instead of Rusanov fluxes used in the present work. This of course requires using a suitable variable, as achieved for instance in22, 12 _{for shallow water}

equations with topograhy and isentropic Euler equations in a porous medium. The main advantage of the latter solver is that its accuracy is increased when compared with the scheme introduced in27_{. Its}

extension to the framework of two-phase two-fluid models is currently in progress and seems feasable, and its counterpart for homogeneous flows also seems rather satisfactory (22_).

• The extension to the framework of three phase flows in a porous medium is also feasable, following23

for instance.

• The extension of the present WBR scheme to the framework of three-dimensional flows is straightfor-ward (see14_{), and it provides satisfactory results, while preserving the basic positivity results given in}

property 8.

Acknowledgments

We would like to thank Fr´ed´eric Archambeau and Thierry Gallou¨et who kindly accepted to read the initial version of the manuscript, which helped much. This work has been achieved in the framework of the NEPTUNE project, with financial support from CEA (Commissariat `a l’Energie Atomique), EDF, IRSN (Institut de Radioprotection et Suret´e Nucl´eaire) and AREVA-NP. Part of the financial support of the first author is provided by ANRT (Association Nationale de la Recherche Technique, Minist`ere de la Recherche) through a EDF/CIFRE contract, and also by FSE (Fonds Social Europ´een). All computational facilities were provided by EDF.

IX.

Appendix A : ”Construction of solutions of the Riemann problem”

We detail here the construction of the solutions used in the main part of the paper. We start with a given state L, and then successively calculate states A, B and R. We assume here that the EOS for each phase is given by:

(γk− 1)ρkek= Pk

A. Given parameters

We assume that the left state (ǫL, α2L, ρ1L, ρ2L, P1L, P2L, U1L, U2L) is given (with (U2)L> 0), and we also

choose ǫR and α2R, so that : ǫLǫR6= 0, α2Lα2R6= 0 and α1Lα1R6= 0.

B. Construction of the first intermediate state A :

We first obviously get :

ǫA= ǫR

Since states L and A are separated by the steady LD wave associated with λ0= 0, their connection is thus

ensured by Riemann invariants I0

n(W ), n = 1, 2, ..., 7.

• Using Riemann invariant I0

1(W ) provides :

α2A= α2L

due to the fact that U2A > 0 (see below).

• Riemann invariants I0

3(W ) and I60(W ) enable to write :

U1A= (ǫL)(ρ1L)(U1L) (ǫA)(ρ1A) ; U2A= (ǫL)(ρ2L)(U2L) (ǫA)(ρ2A)